Mesh Generation Problem Research Articles

3D Bézier Guarding: Boundary-Conforming Curved Tetrahedral Meshing

We present a method for the generation of higher-order tetrahedral meshes. In contrast to previous methods, the curved tetrahedral elements are guaranteed to be free of degeneracies and inversions while conforming exactly to prescribed piecewise polynomial surfaces, such as domain boundaries or material interfaces. Arbitrary polynomial order is supported. Algorithmically, the polynomial input surfaces are first covered by a single layer of carefully constructed curved elements using a recursive refinement procedure that provably avoids degeneracies and inversions. These tetrahedral elements are designed such that the remaining space is bounded piecewise linearly. In this way, our method effectively reduces the curved meshing problem to the classical problem of linear mesh generation (for the remaining space).

A gradient based numerical algorithm to solve inverse dynamic viscoelastic problems of multi-variable identification

A gradient based numerical method is put forward to solve the inverse dynamic viscoelastic problem with multi-variable identification for the constitutive and TKBC (third kind boundary conditions) related parameters. A three parameter model is employed to describe the time dependent TKBC concerned with structural interaction, and is formulated in the forward and inverse dynamic viscoelastic analysis. In addition to the reliable platform provided by TPAA-SBFEM to gain the solution of forward problems, a piecewise recursive adaptive algorithm is developed to obtain the derivatives required in the sensitivity analysis, which is able to provide a stable temporal solution accuracy for different step sizes. SBFEM (Scaled Boundary Finite Element Method) is stressed in the structural analysis with its own advantages, such as the efficient and convenient treatment of stress singularity problem and flexibility of mesh generation with polygonal elements and image based quadtree gridding. The inverse problem of identification is solved by the Levenberg-Marquardt method, and the regional inhomogeneity, crack, initial guess, location/number of measurement points, material contrast ratios, and noisy data are taken into account. Numerical examples are provided to verify the proposed approach, and the satisfactory results are achieved.

Localized MQ-RBF meshless techniques for modeling unsaturated flow

In this study, we focus on space-time mesh-free numerical techniques for efficiently solving the Richards equation which is often used to model unsaturated flow through porous media. We propose an efficient approach which combines the use of local multiquadric (MQ) radial basis function (RBF) methods and space-time techniques. The localized MQ-RBFs meshless methods allow to avoid mesh generation and ill-conditioning problem where a sparse matrix is obtained for the global system which has the advantage of using reduced memory and computational time. To further reduce the computational cost, we use the space-time techniques having the advantages of solving the resulting algebraic system only once and removing the time-integration procedure. The proposed method has the benefit of considering collocation points on the boundaries of computational domains which makes it more flexible in dealing with complex geometries. We implement the proposed numerical model of infiltration and we perform a series of numerical tests, encompassing various nontrivial solutions, to confirm the performance of the proposed techniques. The numerical simulations show the accuracy, efficiency in terms of computational cost, and capability of the proposed numerical techniques in solving the Richards equation in two-, three- and four-dimensional space-time domains with complex boundaries.

A conservative, consistent, and scalable meshfree mimetic method

Mimetic methods discretize divergence by restricting the Gauss theorem to mesh cells. Because point clouds lack such geometric entities, construction of a compatible meshfree divergence remains a challenge. In this work, we define an abstract Meshfree Mimetic Divergence (MMD) operator on point clouds by contraction of field and virtual face moments. This MMD satisfies a discrete divergence theorem, provides a discrete local conservation principle, and is first-order accurate. We consider two MMD instantiations. The first one assumes a background mesh and uses generalized moving least squares (GMLS) to obtain the necessary field and face moments. This MMD instance is appropriate for settings where a mesh is available but its quality is insufficient for a robust and accurate mesh-based discretization. The second MMD operator retains the GMLS field moments but defines virtual face moments using computationally efficient weighted graph-Laplacian equations. This MMD instance does not require a background grid and is appropriate for applications where mesh generation creates a computational bottleneck. It allows one to trade an expensive mesh generation problem for a scalable algebraic one, without sacrificing compatibility with the divergence operator. We demonstrate the approach by using the MMD operator to obtain a virtual finite-volume discretization of conservation laws on point clouds. Numerical results in the paper confirm the mimetic properties of the method and show that it behaves similarly to standard finite volume methods.

A simple diffuse interface approach on adaptive Cartesian grids for the linear elastic wave equations with complex topography

In most classical approaches of computational geophysics for seismic wave propagation problems, complex surface topography is either accounted for by boundary-fitted unstructured meshes, or, where possible, by mapping the complex computational domain from physical space to a topologically simple domain in a reference coordinate system. However, all these conventional approaches face problems if the geometry of the problem becomes sufficiently complex. They either need a mesh generator to create unstructured boundary-fitted grids, which can become quite difficult and may require a lot of manual user interactions in order to obtain a high quality mesh, or they need the explicit computation of an appropriate mapping function from physical to reference coordinates. For sufficiently complex geometries such mappings may either not exist or their Jacobian could become close to singular. Furthermore, in both conventional approaches low quality grids will always lead to very small time steps due to the Courant-Friedrichs-Lewy (CFL) condition for explicit schemes. In this paper, we propose a completely different strategy that follows the ideas of the successful family of high resolution shock-capturing schemes, where discontinuities can actually be resolved anywhere on the grid, without having to fit them exactly. We address the problem of geometrically complex free surface boundary conditions for seismic wave propagation problems with a novel diffuse interface method (DIM) on adaptive Cartesian meshes (AMR) that consists in the introduction of a characteristic function 0≤α≤1 which identifies the location of the solid medium and the surrounding air (or vacuum) and thus implicitly defines the location of the free surface boundary. Physically, α represents the volume fraction of the solid medium present in a control volume. Our new approach completely avoids the problem of mesh generation, since all that is needed for the definition of the complex surface topography is to set a scalar color function to unity inside the regions covered by the solid and to zero outside. The governing equations are derived from ideas typically used in the mathematical description of compressible multiphase flows. An analysis of the eigenvalues of the PDE system shows that the complexity of the geometry has no influence on the admissible time step size due to the CFL condition. The model reduces to the classical linear elasticity equations inside the solid medium where the gradients of α are zero, while in the diffuse interface zone at the free surface boundary the governing PDE system becomes nonlinear. We can prove that the solution of the Riemann problem with arbitrary data and a jump in α from unity to zero yields a Godunov-state at the interface that satisfies the free-surface boundary condition exactly, i.e. the normal stress components vanish. In the general case of an interface that is not aligned with the grid and which is not infinitely thin, a systematic study on the distribution of the volume fraction function inside the interface and the sensitivity with respect to the thickness of the diffuse interface layer has been carried out. In order to reduce numerical dissipation, we use high order discontinuous Galerkin (DG) finite element schemes on adaptive AMR grids together with a second order accurate high resolution shock capturing subcell finite volume (FV) limiter in the diffuse interface region. We furthermore employ a little dissipative HLLEM Riemann solver, which is able to resolve the steady contact discontinuity associated with the volume fraction function and the spatially variable material parameters exactly. While the method is locally high order accurate in the regions without limiter, the global order of accuracy of the scheme is at most two if the limiter is activated. It is locally of order one inside the diffuse interface region, which is typical for shock-capturing schemes at shocks and contact discontinuities. We show a large set of computational results in two and three space dimensions involving complex geometries where the physical interface is not aligned with the grid or where it is even moving. For all test cases we provide a quantitative comparison with classical approaches based on boundary-fitted unstructured meshes.

Unsteady advection-diffusion simulations in complex Discrete Fracture Networks with an optimization approach

It is widely recognized that the prediction of transport of contaminants in a fractured rock mass requires models that preserve several distinctive features of the inner fracture network, like heterogeneity and directionality; in this respect, Discrete Fracture Networks (DFNs) play a significant role. The solution of the associated equations would claim a high computational demand, that could be met only by using agile and robust numerical techniques. In this note a new numerical technique, fully validated from a mathematical standpoint, is applied to engineering problems, also introducing dispersion models for the description of non-stationary transport phenomena. The method results in a fast and scalable resolution tool, based on a PDE-constrained optimization approach designed to avoid mesh generation problems and allowing for transport simulations with an Eulerian approach. Examples are reported to show the quality of the solution obtained, even by using relatively coarse meshes and quite geometrically complex DFNs.

A spring system method for a mesh generation problem

A new direct method for the 2d-mesh generation for a simply-connected domain using a spring system is observed. The method can be used with other methods to modify a mesh for growing solid problems. Advantages and disadvantages of the method are shown. Different types of boundary conditions are explored. The results of modelling for different target domains are given. Some applications for composite materials are studied.

Anisotropic geometry-conforming d-simplicial meshing via isometric embeddings

We develop a dimension-independent, Delaunay-based anisotropic mesh generation algorithm suitable for integration with adaptive numerical solvers. As such, the mesh produced by our algorithm conforms to an anisotropic metric prescribed by the solver as well as the domain geometry, given as a piecewise smooth complex. Motivated by the work of Lévy and Dassi [10-12,20], we use a discrete manifold embedding algorithm to transform the anisotropic problem to a uniform one. This work differs from previous approaches in several ways. First, the embedding algorithm is driven by a Riemannian metric field instead of the Gauss map, lending itself to general anisotropic mesh generation problems. Second we describe our method for computing restricted Voronoi diagrams in a dimension-independent manner which is used to compute constrained centroidal Voronoi tessellations. In particular, we compute restricted Voronoi simplices using exact arithmetic and use data structures based on convex polytope theory. Finally, since adaptive solvers require geometry-conforming meshes, we offer a Steiner vertex insertion algorithm for ensuring the extracted dual Delaunay triangulation is homeomorphic to the input geometries.The two major contributions of this paper are: a method for isometrically embedding arbitrary mesh-metric pairs in higher dimensional Euclidean spaces and a dimension-independent vertex insertion algorithm for producing geometry-conforming Delaunay meshes. The former is demonstrated on a two-dimensional anisotropic problem whereas the latter is demonstrated on both 3d and 4d problems.

An adaptive, residual based, splitting approach for the penalized Navier Stokes equations

The interest on Immersed Boundary Methods (IBM) is increasing in Computational Fluid Dynamics as they simplify the mesh generation problem. In this work, we consider an approach based on the addition of a penalty term to the Navier–Stokes equations to account for the wall boundary conditions. To discretize the resulting equations we use a residual distribution approach previously developed by some of the authors. To adapt the method to the IBM considered, we developed a new formulation of residual distribution based on a Strang splitting method in time, coupling an implicit asymptotic integration procedure of the penalization ODE with a simplified explicit residual distribution for the Navier–Stokes equations. The first method, provides an operator which is exact up to orders η2, with η the penalty parameter assuming values of the order of 10−10. A modification of the solution gradient reconstruction necessary for the evaluation of the viscous fluxes, is also introduced in the paper. This guarantees that correct physical values of the viscous stresses are recovered in vicinity of the solid. We show formally and numerically that the approach proposed is second order accurate for smooth solutions. We evaluate its potential for IBM by coupling the resulting method with unstructured mesh adaptation on wall boundaries. Several steady and time dependent tests are used to show the promising features of the method proposed.

Towards effective flow simulations in realistic discrete fracture networks

We focus on the simulation of underground flow in fractured media, modeled by means of Discrete Fracture Networks. Focusing on a new recent numerical approach proposed by the authors for tackling the problem avoiding mesh generation problems, we further improve the new family of methods making a step further towards effective simulations of large, multi-scale, heterogeneous networks. Namely, we tackle the imposition of Dirichlet boundary conditions in weak form, in such a way that geometrical complexity of the DFN is not an issue; we effectively solve DFN problems with fracture transmissivities spanning many orders of magnitude and approaching zero; furthermore, we address several numerical issues for improving the numerical solution also in quite challenging networks.

TetGen, a Delaunay-Based Quality Tetrahedral Mesh Generator

TetGen is a C++ program for generating good quality tetrahedral meshes aimed to support numerical methods and scientific computing. The problem of quality tetrahedral mesh generation is challenged by many theoretical and practical issues. TetGen uses Delaunay-based algorithms which have theoretical guarantee of correctness. It can robustly handle arbitrary complex 3D geometries and is fast in practice. The source code of TetGen is freely available. This article presents the essential algorithms and techniques used to develop TetGen. The intended audience are researchers or developers in mesh generation or other related areas. It describes the key software components of TetGen, including an efficient tetrahedral mesh data structure, a set of enhanced local mesh operations (combination of flips and edge removal), and filtered exact geometric predicates. The essential algorithms include incremental Delaunay algorithms for inserting vertices, constrained Delaunay algorithms for inserting constraints (edges and triangles), a new edge recovery algorithm for recovering constraints, and a new constrained Delaunay refinement algorithm for adaptive quality tetrahedral mesh generation. Experimental examples as well as comparisons with other softwares are presented.

BACKGROUND GRID METHOD FOR PLANE SHAPES TRIANGULATION IN FUNCTIONAL APPROACH

In herein paper is described the problem of triangular mesh generation for complex geometrical domain. Functional approach is used for a geometrical object representation. Vladimir Rvachev’s R functions are used also. Functional approach is based on implicit functions and logical operations (negation, conjunction and disjunction) over these functions. Logical operation is a special real-value function that at inner point is greater than zero and at outer point is less than zero. Thus standard triangulations techniques, which based on predefined boundary discreet model, is less efficient. During background grid method for triangulation, triangles are built in domain directly. Presented method starts with a mesh that can be relatively easy generated (e.g. uniform mesh). Next, initial mesh is defined as a set of all inner elements. The last one generates a layer of elements near boundary (adaptation step). Adaptive method generates meshes that are close to uniform for structured background grids.

Special Aspects of Wind Wave Simulations for Surge Flood Forecasting and Prevention

The paper is focused on several issues of wind wave simulations with SWAN model for the tasks related to prevention of surge floods in St. Petersburg. It introduces main objectives that are pursued through the use of the model as well as covers problems of computational mesh generation and model parameter calibration. We also examined several assumptions on the necessity to take ice fraction and sea level rise into account in wind wave simulations.

Boundary recovery for 3D Delaunay triangulation

New ideas are presented in this paper for the boundary recovery of 3D Delaunay triangulation. Fully constrained Delaunay triangulations in terms of geometrical and topological integrities on all boundary edges and facets are required in many applications, such as meshing by components, fluid–structure interactions, parallel mesh generation, local remeshing and interface problems. The geometry of boundary edges and facets can be recovered by the introduction of Steiner points. However, for a fully constrained Delaunay triangulation, these Steiner points have to be removed or repositioned towards the interior of the domain to restore the topological integrity of the boundary edges and the facets. It is found that Steiner points on edges could be removed more systematically following a specific sequence in an alternative manner rather than a random selection commonly adopted in practice; whereas for Steiner points on a facet, a weight on the Steiner point adjacency would lead to an optimal order to facilitate their removal. A linear programming technique is also employed to determine the feasible region for the relocation of Steiner points in the interior of the domain. Work examples and industrial applications with details in the boundary recovery are presented to illustrate how the algorithm works on objects with difficult boundary conditions.

Towards the Integration of Topology Optimization into the CAD Process

This paper presents a contribution to the automation and integration of topology optimization methods (TOM) with CAD, in the context of the design of statically loaded mechanical structures and parts. Starting from an initial CAD model with relevant engineering data, the goal is automatically generating an optimized CAD model with respect to engineering objectives and constrains. Though many optimization methods are now available, their complete and efficient integration into the design process faces several problems. After introducing the basic steps involved in the whole process and identifying the challenges inherent to this integration, this paper presents our contribution in addressing these challenges. The paper is focused on the specification of design and non-design sub-domains, on automatic mesh generation problems induced and on the adaptation of TO concepts in the context of 3D unstructured meshes. TO itself is adapted from a SIMP scheme, which is an arbitrary choice as any other optimization m...

An immersed boundary method using unstructured anisotropic mesh adaptation combined with level-sets and penalization techniques

The interest on embedded boundary methods increases in Computational Fluid Dynamics (CFD) because they simplify the mesh generation problem in the case of the Navier–Stokes equations. The same simplifications occur for the simulation of multi-physics flows, the coupling of fluid–solid interactions in situation of large motions or deformations, to give a few examples. Nevertheless an accurate treatment of the wall boundary conditions remains an issue of the method. In this work, the wall boundary conditions are easily taken into account through a penalization technique, and the accuracy of the method is recovered using mesh adaptation, thanks to the potential of unstructured meshes. Several classical examples are used to demonstrate that claim.

Reuse of Digital Mock-Up Data by Using Shape Matching

We present a framework for reuse of digital mock-up models. Our idea is to find reusable models in the archives by shape matching. Since the reusable models are already designed by operators in the past projects, we can easily create various types of data by using existing models. Indeed, we demonstrate this framework is efficient for surface simplification and CAE mesh generation problems.

Conservative unsteady aerodynamic simulation of arbitrary boundary motion using structured and unstructured meshes in time

SUMMARYSimulation of unsteady fluid behaviour with arbitrary boundary motion or topological change remains restricted owing to mesh deformation limitations, and usually requires the application of special techniques using overlapping meshes, sliding planes, remeshing or immersed boundaries. This work presents the application of a spacetime interpretation of the fluid conservation laws that unifies meshes in space and time. This effectively replaces the problem of mesh deformation with the problem of mesh generation and, because connectivity is no longer restricted to being constant in time, any motion or topological change may be simulated. Examples are given applying the method to a piston, a pitching NACA0012 aerofoil, an appearing/disappearing object, a two‐dimensional store separation and a rotation case to validate and then demonstrate the capabilities of the method. Results for the pitching aerofoil case are compared with a conventional moving mesh unsteady computation and shown to be consistent, whereas the demonstration cases show qualitatively correct behaviour and illustrate the general nature of the technique. Copyright © 2011 John Wiley & Sons, Ltd.

Tailoring unstructured meshes for use with a 3D time domain co‐volume algorithm for computational electromagnetics

AbstractThe Yee algorithm is a highly efficient time domain co‐volume method for computational electromagnetics using a pair of staggered orthogonal cartesian meshes. An equivalent unstructured mesh process is readily implemented on a primal Delaunay tetrahedral mesh and its orthogonal Voronoï dual. The difficulty, for practical applications, is ensuring that both the Delaunay and the Voronoï meshes possess the necessary properties to enable an accurate and efficient solution to be obtained by this approach. This is a mesh generation problem and it is addressed here by using a combination of a point distribution provided by an ideal tetrahedral mesh, a constrained centroidal Voronoï tessellation method and a mesh quality optimization procedure. For the selected application area of electromagnetic wave scattering problems, the computational efficiency is enhanced by using a hybrid primal mesh, consisting of tetrahedral elements in the vicinity of the scatterer and hexahedral elements elsewhere. Three examples are included to demonstrate the viability of the proposed approach and to indicate the numerical performance that can be achieved. Copyright © 2010 John Wiley & Sons, Ltd.

Density Functions for Mesh Refinement in Numerical Optimal Control

needed to understand how the density/monitor functions can be used in numerical optimal control and how they can influence the accuracy and robustness of numerical optimal control algorithms. Furthermore, the choice of “good” density/monitor functions for mesh discretization of optimal control problems seems to be open. Inthispaperweattempttoprovideapartialanswertotheprevious questions. We introduce a method to distribute the mesh points efficiently using density/monitor functions. Although different monitor functions can be used for mesh generation, an appropriate choice of a monitor function can generate a better quality mesh, and can improve the accuracy of the solution along with the speed of convergence. Hence, the problem of mesh generation can be treated asaproblemof findinganappropriatedensity/monitorfunction.Two possible choices of density functions are used in the numerical examples, based on the discrete control/state histories from the previousiterationduringthemeshrefinementprocess.Theproposed method avoids the numerical integration step and the use of ODE solversforthesystemdynamicsaswasdonein[8].Yet,itgeneratesa mesh with a suitable level of adaptive discretization that provides sharp resolution around the places where the control switches or the trajectory meets/leaves state constraints, thus resulting in better accuracy of the overall final solution. Numerical examples are presented to demonstrate the advantage of the proposed method, and comparisons are provided against the industry-standard sparse optimal control software (SOCS).